
\(K_{{{G}}}+ f^*(K_{{F}}+\Delta )\) ,

\(Y\)  is smooth,

\({{G}}\)  has only reduced singularities,

\( has the simple normal crossing support,\item the proper transform \) Y\( on \) Y\( of \)  is smooth,

the union of the support of \( and the exceptional divisor of \) f\( is \) iTi\(,\item all non-invariant irreducible components of \) iTi\( are pairwisely disjoint, and\item any separatrix \) C\( through a non-smooth foliation point on a non-invariant prime divisor \) Ti\( has \) CTi=1\(.\) 
Then we have
\(\textnormal {mld}({{F}},=\min \left\lbrace \begin{array}{l}\min _{E_i\in \widetilde{I}}\lbrace 1+a(E_i,{{F}},\Delta )\rbrace ,\min _{E_i\notin \widetilde{I}}\lbrace a(E_i,{{F}},\Delta )\rbrace , \\\min _{(f_*)^{-1}D_i\in \widetilde{I}}\lbrace 1-a_i\rbrace ,\min _{(f_*)^{-1}D_i\notin \widetilde{I}}\lbrace -a_i\rbrace , 1\end{array}\right\rbrace \) 

where \(\widetilde{I} = \lbrace T_k\,\vert \, T_k \mbox{ in non-invariant and has only smooth foliation points}\rbrace \) .
Proof First we take \(\pi : (Y,{{F}}_Y) \rightarrow (X,{{F}})\)  to be the minimal resolution of \((X,{{F}})\) .
Then we take \(\phi : Z \rightarrow Y\)  to be a log resolution of \((Y,(\pi _*)^{-1}\Delta +\textnormal {Exc}(\pi ))\) .
After some further blowups, this gives the existence of such \(f\) .
Let \(b_j = -a(E_j,{{F}},\Delta )\le \varepsilon (E_j)\) .
By Proposition REF , we have that
\(\textnormal {mld}({{G}},\)  is a minimum of numbers of the following forms:

1.

\(-a_i\)  if \((f_*)^{-1}D_i\notin \widetilde{I}\) .

\(1-a_i\)  if \((f_*)^{-1}D_i\in \widetilde{I}\) .

\(-b_j\)  if \(E_j\notin \widetilde{I}\) .

\(1-b_j\)  if \(E_j\in \widetilde{I}\) .

\(-a_i-b_j\)  where \((f_*)^{-1}D_i\cap E_j\ne \emptyset \)  has some non-smooth foliation singularities.

\(1-a_i-b_j\)  where \((f_*)^{-1}D_i\cap E_j\ne \emptyset \)  has only smooth foliation singularities.

\(-b_{j_1}-b_{j_2}\)  for \(E_{j_1}\cap E_{j_2}\ne \emptyset \)  has some non-smooth foliation singularities.

\(1-b_{j_1}-b_{j_2}\)  for \(E_{j_1}\cap E_{j_2}\ne \emptyset \)  has only smooth foliation singularities.

Then it suffices to show the following claim.
Claim The last four terms above are irrelevant when taking the minimum.

Proof (Claim) 
It is clear if one of \(a_i\)  and \(b_j\)  (resp. \(b_{j_1}\)  and \(b_{j_2}\) ) is non-positive.
So we may assume that both numbers are positive.
Moreover, the associated divisors are contained in \(\widetilde{I}\) ; otherwise, either \(-a_i\)  or \(-b_j\)  (resp. either \(-b_{j_1}\)  or \(-b_{j_2}\) ) is strictly less than zero, which gives a contradiction.
Thus, both associated divisors are non-invariant, and hence they have empty intersections.
This completes the proof of the claim.

\(\Box \)

Ascending chain condition for foliated minimal log discrepancy
In this section, we fix the following notations.
The set \(B\subset [0,1]\)  is always assumed to satisfy the descending chain condition.
Let \(({{F}},x)\)  be the germ of a log canonical foliation singularity where \( is a \) Q\(-divisor whose coefficients are in \) B\(.Let \) ={Ej}j=1r\( be the dual graph of exceptional divisors of a resolution \) : (Y,G) (X,F,x)\( and \)  i=1s biBi\( be the proper transform of \)  on \(Y\) .
Let \(a_j := a(E_j,{{F}},\ge 0\)  all \(j=1\) , \(\ldots \) , \(r\) .
Lemma 4.1 
Assume that \(\Gamma \)  is a tree, all \(E_j\)  are invariant, and \(a_j\le 1\)  for all \(j=1\) , \(\ldots \) , \(r\) .
Then we have:

If \(a_j \ge \varepsilon \)  for some positive real number \(\varepsilon \) , then \(-E_j^2 \le \lfloor \frac{2}{\varepsilon }\rfloor \) .

If \(-E_j^2\ge 2\)  for some \(j\) , then
\(1-a_j \ge \frac{1-a_k}{2}\) 
and \(2a_j\le a_k + a_\ell \)  for any \(k\ne \ell \)  such that \(E_j\cdot E_k = E_j\cdot E_\ell = 1\) .

Suppose the vertex \([E_{j_0}]\)  is a fork and the vertices \([E_{j_k}]\)  are connected to \([E_{j_0}]\)  for \(k=1\) , 2, 3.
If \(-E_{j_k}^2\ge 2\)  for \(k=0\) , 1, 2, then \(a_{j_0}\le a_{j_3}\) .

If \(-E_j^2=1\)  for some \(j\) , \(-E_k^2\ge 2\) , \(a_j<a_k\)  for all \(k\ne j\) , and the vertex \([E_j]\)  is not a fork, then \(\Gamma \)  is a chain.

Given a sequence of vertices \([E_1]\) , \(\ldots \) , \([E_m]\)  where \(E_i\cdot E_{i+1}=1\)  for all \(i=1\) , \(\ldots \) , \(m-1\) .
Suppose \(-E_i^2\ge 2\)  for \(i = 2\) , \(\ldots \) , \(m-1\)  and \(a_1\le a_2\) .
If either \(-E_2^2\ge 3\)  and \(a_2\ge \varepsilon \)  or \(E_2\ge \varepsilon \) , then \(m\le \lfloor \frac{1}{\varepsilon }\rfloor + 2\) .


Proof Notice that
\(\left\lbrace \begin{array}{l}\begin{aligned}0 &= \left(K_{{G}}+ \sum _{i=1}^r a_iE_i\right)\cdot E_j \\&= K_{{G}}\cdot E_j + \sum _{i=1}^s t_{i,j}b_i - a_jE_j^2 - \sum _{E_i\cdot E_j=1} a_i \\&\ge \#\lbrace i\,\vert \, E_i\cdot E_j=1\rbrace  -2 + \sum _{i=1}^s t_{i,j}b_i - a_jE_j^2 - \sum _{E_i\cdot E_j=1} a_i \\&= -2 + \sum _{i=1}^s t_{i,j}b_i - a_jE_j^2 + \sum _{E_i\cdot E_j=1}(1-a_i)\end{aligned}\end{array}\right.\) 
where \(t_{i,j} =B_i\cdot E_j\ge 0\) .

From the inequality (REF ), we have that \(0\ge -2 - a_jE_j^2\) .
Thus,
\(-E_j^2 \le \frac{2}{a_j} \le \frac{2}{\varepsilon }.\) 
Since \(-E_j^2\in {\mathbb {Z}}\) , we have that \(-E_j^2 \le \lfloor \frac{2}{\varepsilon }\rfloor \) .

From the inequality (REF ), we have that \(0\ge -2 - a_jE_j^2 + 1 - a_k\ge -1+2a_j-a_k\)  and \(0\ge -2 - a_jE_j^2 + 1 - a_k+1-a_\ell \ge 2a_j-a_k-a_\ell \) .
Then we get \(1-a_j \ge \frac{1-a_k}{2}\)  and \(2a_j\le a_k + a_\ell \) .

By (b), we have
\(1-a_{j_k}\ge \frac{1-a_{j_0}}{2}\) 
for \(k=1\) , 2. Thus
\(1-a_{j_1}+1-a_{j_2} \ge 1-a_{j_0}.\) 
Also from the inequality (REF ), we have
\(0\ge 1 +2a_{j_0} - \sum _{k=1}^3 a_{j_k}. \) 
Combining with the inequality (REF ), we get
\(a_{j_3}\ge 1+2a_{j_0}-a_{j_1}-a_{j_2}\ge 2a_{j_0}-a_{j_0} = a_{j_0}.\) 

Assume \(\Gamma \)  is not a chain.
Let the vertex \([E_k]\)  be a fork.
Let \([E_k] =: [E_{\ell _0}]\) , \([E_{\ell _1}]\) , \(\ldots \) , \([E_{\ell _m}] := [E_j]\)  be a sequence of vertex connecting from \([E_k]\)  to \([E_j]\) .
By (c), we know that \(a_{\ell _0}\le a_{\ell _1}\) .
By (b), we get \(a_{\ell _1}\le a_{\ell _2} \le \ldots \le a_{\ell _m}\) .
Thus, \(a_k\le a_j\) , which gives a contradiction.

From the inequality (REF ) with \(j=2\) , we have that
\(0 &\ge & -2+\sum _{i=1}^s t_{i,2}b_i-a_2E_2^2 + (1-a_1)+(1-a_3) \\&\ge & 2a_2-a_1-a_3+\varepsilon \\&\ge & a_2-a_3+\varepsilon \) 
Therefore, \(a_3\ge a_2+\varepsilon \) .
Claim \(a_{j+1}-a_j \ge \varepsilon \)  for all \(j = 2\) , \(\ldots \) , \(m-1\) .

Proof (Claim) 
We have seen the claim holds true when \(j=2\) .
Then by (b), we have \(a_{j+1}+a_{j-1}\ge 2a_j\) .
Thus, \(a_{j+1}-a_j\ge a_j-a_{j-1}\ge \varepsilon \)  by induction on \(j\) .

Therefore, we have
\(1\ge a_m \ge a_{m-1}+\varepsilon \ge \cdots \ge a_2+(m-2)\varepsilon \ge (m-2)\varepsilon .\) 
Hence, \(m\le \lfloor \frac{1}{\varepsilon }\rfloor + 2\) .

\(\Box \)

Lemma 4.2 
Assume that \((Y,{{G}})\)  is the minimal resolution of the germ \((X,{{F}},x)\) . Then we have the followings:

\(\textnormal {pld}_x({{F}}) = 0\)  if \(\Gamma \)  is not of the type (1) in Theorem REF .

If \(a_j>0\)  for all \(j\) , then \(a_j \le a(E_j,{{F}})\le 1\)  for all \(E_j\) 's.

Fix a number \(\delta \) , there are only finitely many sequences \(\lbrace t_1\) , \(\ldots \) , \(t_r\rbrace \)  where \(t_i\in {\mathbb {N}}\)  such that \(\sum _{i=1}^r t_ib_i \le \delta \)  for some \(b_i\in B\backslash \lbrace 0\rbrace \) .


Proof

This is straightforward.

Note that
\(\left(\sum _{i=1}^r a_iE_i\right)\cdot E_j = (K_{{G}}+\cdot E_j\ge K_{{G}}\cdot E_j = \left(\sum _{i=1}^r a(E_i,{{F}})E_i\right)\cdot E_j\) 
for all \(j\) .
By Lemma REF , we have \(0<a_j \le a(E_j,{{F}})\)  for all \(j\) .
Then \(\Gamma \)  is of type (1) in Theorem REF .
Also, we notice that
\(K_{{G}}\cdot E_j\ge \left(\sum _{i=1}^r E_i\right)\cdot E_j.\) 
Thus, we have \(a(E_j,{{F}})\le 1\)  by Lemma REF .

Since \(B\)  is a DCC set, there is a positive number \(\varepsilon \)  such that \(b_i\ge \varepsilon \)  for all \(b_i\in B\backslash \lbrace 0\rbrace \) .
Note that
\(\delta \ge \sum _{i=1}^r t_ib_i \ge \sum _{i=1}^r t_i\varepsilon \mbox{ and thus } \sum _{i=1}^r t_i\le \frac{\delta }{\varepsilon }.\) 
This shows that there are only finitely many possible \(r\)  and, for any fixed \(r\) , there are only finitely many sequences \(\lbrace t_1\) , \(\ldots \) , \(t_r\rbrace \)  such that \(\sum t_i\le \frac{\delta }{\varepsilon }\) .
This proves (c).

\(\Box \)

Lemma 4.3 
Fix \(\varepsilon >0\) .
Suppose \((Y,{{G}})\)  is the minimal resolution of the germ \((X,{{F}},x)\)  with \(\textnormal {pld}_x({{F}},\ge \varepsilon \)  and \(b_j\ge \varepsilon \)  for all \(j\) .
Then \(\Gamma \)  belongs to one of the following cases:

Finitely many graphs (that include the way how \(B_i\)  intersects \(E_j\) ).

The chain \(\cup _jE_j\)  given by the ordered curves \(L_{\ell _1}\) , \(\ldots \) , \(L_1\) , \(M_1\) , \(\ldots \) , \(M_n\) , \(R_1\) , \(\ldots \) , \(R_{\ell _2}\)  where \(K_{{G}}\cdot L_{\ell _1}=-1\) , the weights of \(M_k\) 's are 2, and each \(B_j\)  does not meet any \(M_k\) 's.
Moreover, the partial log discrepancy \(\textnormal {pld}_x({{F}},\)  is achieved at either \(L_1\)  or \(R_1\)  and there are only finitely many possibilities (independent of \(n\) ) for the dual graphs \(\lbrace L_1\) , \(\ldots \) , \(L_{\ell _1}\rbrace \)  and \(\lbrace R_1\) , \(\ldots \) , \(R_{\ell _2}\rbrace \)  and the way how \(B_i\)  intersects \(L_\alpha \)  and \(R_\beta \) .


Proof By Lemma REF , we have that \(\Gamma \)  is of type (1), and therefore \(\Gamma \)  satisfies the assumption in Lemma REF .
So from the inequality (REF ), we have that \(0\ge -2 + \sum t_{i,j}b_i -a_jE_j^2\)  and hence \(\sum t_{i,j}b_i\le 2+a_jE_j^2\le 2\) .
Therefore, there are only finitely many possibilities for \(t_{i,j}\)  for any \(i\) , \(j\)  by Lemma REF  (c).
Let the chain be \(\cup _jE_j\) .
By Lemma REF  (b), the function \(f: \lbrace 1\) , \(\ldots \) , \(r\rbrace  \rightarrow {\mathbb {R}}\)  which maps \(j\)  to \(a_j\)  is convex.
Let \(S = \lbrace j\vert \, a_j=\min _k\lbrace a_k\rbrace \rbrace \) , \(N=\#S\) , and \(j_0=\min S\) .
If \(N\ge 2\) , then \(S = \lbrace j_0\) , \(\ldots \) , \(j_0+N-1\rbrace \)  by convexity of \(f\) .
Note that, by inequality (REF ) with fixed \(j\) , we have
\(0\ge -2 + \sum _{i=1}^s t_{i,j}b_i - a_jE_j^2 + (1-a_{j-1})+(1-a_{j+1})\) 
and thus
\(a_{j-1}-2a_j+a_{j+1}\ge \sum _{i=1}^s t_{i,j}b_i-a_j(E_j^2+2)\ge \left(\sum _{i=1}^s t_{i,j}-E_j^2-2\right)\varepsilon \ge \varepsilon \) 
whenever \(-E_j^2\ge 3\)  or \(t_{i,j}\ge 1\)  for some \(i\) .
So we can choose \(L_1 = E_{j_0}\)  and \(R_1 = E_{j_0+N-1}\) .
By Lemma REF  (e), we have \(\ell _i\le \lfloor \frac{2}{\varepsilon }\rfloor +1\)  for \(i=1\) , 2.
If \(N=1\) , then \(S = \lbrace j_0\rbrace \) .
Let \(T = \lbrace j\vert \, E_j^2=-2 \mbox{ and } E_j\cdot 0\rbrace \) .
We have the following three cases:

If both \(j_0-1\)  and \(j_0+1\)  are not in \(T\) , then we choose \(L_1 = E_{j_0}\)  and \(R_1 = E_{j_0+1}\) .

If \(j_0-1\in T\) , then we choose \(R_1 = E_{j_0}\)  and \(L_1 = E_u\)  where \(u\)  is the maximal integer strictly less than \(j_0\)  and not in \(T\) .

If \(j_0+1\in T\) , then we choose \(L_1 = E_{j_0}\)  and \(R_1 = E_u\)  where \(u\)  is the minimal integer strictly greater than \(j_0\)  and not in \(T\) .

By Lemma REF  (e), we also have \(\ell _i\le \lfloor \frac{2}{\varepsilon }\rfloor +1\)  for \(i=1\) , 2.
To complete the proof, it remains to show the following claim.
Claim There are only finitely many possibilities for the dual graphs \(\lbrace L_1\) , \(\ldots \) , \(L_{\ell _1}\rbrace \)  and \(\lbrace R_1\) , \(\ldots \) , \(R_{\ell _2}\rbrace \)  and the way how \(B_i\)  intersects \(L_\alpha \)  and \(R_\beta \) .

Proof To simplify some notations, we assume that we have a chain \(C = \cup _{j=1}^\ell C_j\)  with increasing associated discrepancies \(a_j\ge \varepsilon \) .
By inequality (REF ), we have
\(1 &\ge & a_\ell \\&\ge & a_{\ell -1} + \left(\sum _{i=1}^s t_{i,\ell -1}-E_{\ell -1}^2-2\right)\varepsilon \\&\vdots & \\&\ge & a_1 + \sum _{j=1}^{\ell -1}\left(\sum _{i=1}^s t_{i,j}-E_{j}^2-2\right)\varepsilon \\&\ge & \varepsilon + \sum _{j=1}^{\ell -1}\left(\sum _{i=1}^s t_{i,j}-E_{j}^2-2\right)\varepsilon .\) 
Thus, we have
\(\sum _{j=1}^{\ell -1}\left(\sum _{i=1}^s t_{i,j}-E_{j}^2-2\right)\le \frac{1}{\varepsilon }-1.\) 
Therefore, there are only finitely many possibilities for \(E_j^2\)  for any \(j\)  since we have seen that there are only finitely many possibilities for \(t_{i,j}\)  for any \(i\) , \(j\) .

\(\Box \)

Lemma 4.4 
For the second case in Lemma REF , if the chain has length \(r = \ell _1+n+\ell _2 > 2n_0+2\) 
where \(n_0 = \lfloor \frac{1}{\varepsilon }\rfloor \) ,
then
\(\textnormal {mld}_x({{F}},\Delta ) = \textnormal {pld}_x({{F}},\Delta ).\) 
Moreover, for the fixed graphs \(\lbrace L_1\) , \(\ldots \) , \(L_{\ell _1}\rbrace \)  and \(\lbrace R_1\) , \(\ldots \) , \(R_{\ell _2}\rbrace \) , the fixed number \(t\)  of irreducible components of \(B\) , and the fixed way how \(B_i\)  intersects \(E_j\) , if \(\lim _{n\rightarrow \infty }\, b_j = \overline{b_j}\)  exists where \((b_1\) , \(\ldots \) , \(b_s)\)  is some ordering of coefficients of \(, then\)\(\textnormal {pld}({{F}},\Delta )\ge \min \left\lbrace \frac{\alpha ^L}{m_1-q_1},\frac{\alpha ^R}{m_2-q_2}\right\rbrace \)\( and\)\(\lim _{n\rightarrow \infty }\,\textnormal {pld}({{F}},\Delta ) = \min \left\lbrace \frac{\overline{\alpha ^L}}{m_1-q_1}, \frac{\overline{\alpha ^R}}{m_2-q_2}\right\rbrace .\)\(where\begin{itemize}\item c_i^L = \sum _j(B_i\cdot L_j)g^L_j and c_i^R = \sum _j(B_i\cdot R_j)g^R_j,\item g^L_j (resp. g^R_j) is the determinant of the chain L_{j+1}, \ldots , L_{\ell _1}(resp. R_{j+1}, \ldots , R_{\ell _2}),\item m_1 = g^L_0, q_1 = g^L_1, m_2 = g^R_0, and q_2 = g^R_1,\item \alpha ^L = 1-\sum _ib_ic_i^L and \alpha ^R = -\sum _ib_ic_i^R,\item \overline{\alpha ^L} = 1-\sum _i\overline{b_i}c_i^L and \overline{\alpha ^R} = -\sum _i\overline{b_i}c_i^R.\end{itemize}\)

Proof We first show that \(\textnormal {mld}_x({{F}},\Delta ) = \textnormal {pld}_x({{F}},\Delta )\)  if \(r> 2n_0+2\) .
Suppose not, then there is an irreducible divisor \(F\)  over \(Y\)  such that \(a(F,{{F}},\Delta )<\textnormal {pld}({{F}},\Delta )\) .
Then \(F\)  is an exceptional divisor for the composition of \(t\)  blowups.
Let \(F_1\) , \(\ldots \) , \(F_t:=F\)  be the exceptional divisors for the corresponding blowups.
We may assume that \(a(F,{{F}},\Delta ) < a(F_j,{{F}},\Delta )\)  for all \(1\le j\le t-1\) .
Note that \(-F_j^2\ge 2\)  for all \(1\le j\le t-1\)  and \([F]\)  is not a fork since \(\sum _{i=1}^{t-1}F_i\)  has the simple normal crossing support.
Then by Lemma REF  (d), the graph \(\lbrace E_i,F_j\rbrace  = \lbrace G_k\rbrace _{k=1}^N\)  is a chain with \(F=F_t=G_u\)  for some \(u\) .
Notice that exactly one of \(-G^2_{u-1}\)  or \(-G^2_{u+1}\)  is 2.
Without loss of generality, we assume that \(-G^2_{u+1} = 2\)  and \(-G^2_{u-1}\ge 3\) .
Then \(u\le n_0+2\)  by Lemma REF  (e).
Put \(\gamma = \min \lbrace k> u\vert -G^2_k\ge 3\rbrace \) .
Then, by Lemma REF  (e) again, we have \(N-\gamma \le n_0\) .
By blowing down \(G_u\) , \(\ldots \) , \(G_{\gamma -1}\)  successively, we get a new foliated surface with only reduced singularities and with at most \((u-1)+(N-\gamma +1)\le 2n_0+2<r\)  exceptional divisors over \(X\) .
However, since this new foliated surface factors through the minimal resolution, the number of exceptional divisors over \(X\)  is at least \(r\) , which gives a contradiction.
Now we compute the partial log discrepancy \(\textnormal {pld}_x({{F}},\Delta )\) .
It is known that the graph \(\Gamma \)  is uniquely determined by these five numbers \(m_1\) , \(q_1\) , \(m_2\) , \(q_2\) , \(n\) .
By Lemma REF , we have that both \(a(L_1,{{F}},\)  and \(a(R_1,{{F}},\)  are between
\(\min \left\lbrace \frac{\alpha ^L}{m_1-q_1},\frac{\alpha ^R}{m_2-q_2}\right\rbrace  \mbox{ and } \max \left\lbrace \frac{\alpha ^L}{m_1-q_1},\frac{\alpha ^R}{m_2-q_2}\right\rbrace .\) 
Moreover, we have that
\(\lim _{n\rightarrow \infty }\,a(L_1,{{F}}, = \frac{\overline{\alpha ^L}}{m_1-q_1} \mbox{ and } \lim _{n\rightarrow \infty }\,a(R_1,{{F}}, = \frac{\overline{\alpha ^R}}{m_2-q_2}.\) 
\(\Box \)

Lemma 4.5 
Notation as in Lemma REF , we have
\(a(L_1,{{F}},\Delta ) &=&\frac{\alpha ^L(n(m_2-q_2)+m_2) + \alpha ^Rq_1}{n(m_1-q_1)(m_2-q_2) + m_2(m_1-q_1) + q_1(m_2-q_2)} \\&=& \frac{\frac{\alpha ^L}{m_1-q_1}(n+\frac{m_2}{m_2-q_2})+\frac{\alpha ^R}{m_2-q_2}\frac{q_1}{m_1-q_1}}{n+\frac{m_2}{m_2-q_2}+\frac{q_1}{m_1-q_1}}\) 
and
\(a(R_1,{{F}},\Delta ) &=&\frac{\alpha ^R(n(m_1-q_1)+m_1) + \alpha ^Lq_2}{n(m_1-q_1)(m_2-q_2) + m_1(m_2-q_2) + q_2(m_1-q_1)} \\&=& \frac{\frac{\alpha ^R}{m_2-q_2}(n+\frac{m_1}{m_1-q_1})+\frac{\alpha ^L}{m_1-q_1}\frac{q_2}{m_2-q_2}}{n+\frac{m_1}{m_1-q_1}+\frac{q_2}{m_2-q_2}}.\) 

Proof We denote \(\Gamma \)  as \(\Gamma _{m_1,q_1,m_2,q_2,n}\)  and its intersection matrix \(A\)  as \(A_{m_1,q_1,m_2,q_2,n}\) .
Let \(S_n = \det (-A_{1,1,m_2,q_2,n})\) .
Then by Lemma REF  (1), we have that
\(\det (-A_{m_1,q_1,m_2,q_2,n}) = 2m_1S_{n-1} - a_1S_{n-1} - m_1S_{n-2} = (2m_1 - q_1)S_{n-1} - m_1S_{n-2}\) 
and \(S_n = 2S_{n-1}-S_{n-2}\) .
Thus, \(S_{i+1}-S_i = S_{i}-S_{i-1}\)  for all \(i\) .
Since \(S_0 = m_2\)  and \(S_1 = 2S_0 - q_2 = 2m_2 - q_2\) , we have
\(S_n = \sum _{i=1}^n(S_i-S_{i-1}) + S_0 = n(S_1-S_0) + S_0 = n(m_2-q_2)+m_2\) 
and thus
\(\det (-A) &=& \det (-A_{m_1,q_1,m_2,q_2,n}) \\&=& (2m_1-q_1)\big ((n-1)(m_2-q_2)+m_2\big ) - m_1\big ((n-2)(m_2-q_2)+m_2\big ) \\&=& n(m_1-q_1)(m_2-q_2) + m_2(m_1-q_1) + q_1(m_2-q_2).\) 
Also by Lemma REF  (2), we have
\((L_1,L_j)\textnormal {-cofactor} &=& (-1)^{|\Gamma |+1} g^L_jS_n, \\(L_1,M_j)\textnormal {-cofactor} &=& (-1)^{|\Gamma |+1} q_1S_{n-j}, \mbox{ and} \\(L_1,R_j)\textnormal {-cofactor} &=& (-1)^{|\Gamma |+1} g^R_jq_1.\) 
Put \(d_j^L = (K_{{G}}+\cdot L_j\) , \(d_j^R = (K_{{G}}+\cdot R_j\) , and \(d_j^M = (K_{{G}}+\cdot M_j\) .
By assumption, we know that \(d_j^M=0\)  and \(d_j^R = R_j\)  for all \(j\) .
Moreover, \(d_j^L = L_j\)  for \(j\ne \ell _1\)  and \(d_{\ell _1}^L = -1+L_{\ell _1}\) .
Then we have
\(\det (A)a(L_1,{{F}},\Delta ) = \sum _{j=1}^{\ell _1}d_j^L\cdot (L_1,L_j)\textnormal {-cofactor} +\sum _{j=1}^{\ell _2}d_j^R\cdot (L_1,R_j)\textnormal {-cofactor}. \) 
Thus
\(& & -\det (-A)a(L_1,{{F}},\Delta ) \\&=& \sum _{j=1}^{\ell _1}d_j^L g^L_jS_n + \sum _{j=1}^{\ell _2}d_j^Rg^R_jq_1 \\&=& \sum _{j=1}^{\ell _1}\left(\sum _{i=1}^sb_i(B_i\cdot L_j)\right)g^L_jS_n - g^L_{\ell _1}S_n + \sum _{j=1}^{\ell _2}\left(\sum _{i=1}^sb_i(B_i\cdot R_j)\right)g^R_jq_1 \\&=& \left(\sum _{i=1}^sb_ic_i^L-1\right)S_n + \left(\sum _{i=1}^sb_ic_i^R\right)q_1\\&=& -\alpha ^LS_n-\alpha ^Rq_1.\) 
Therefore,
\(a(L_1,{{F}},\Delta ) &=&\frac{\alpha ^L\big (n(m_2-q_2)+m_2\big ) + \alpha ^Rq_1}{n(m_1-q_1)(m_2-q_2) + m_2(m_1-q_1) + q_1(m_2-q_2)} \\&=& \frac{\frac{\alpha ^L}{m_1-q_1}(n+\frac{m_2}{m_2-q_2})+\frac{\alpha ^R}{m_2-q_2}\frac{q_1}{m_1-q_1}}{n+\frac{m_2}{m_2-q_2}+\frac{q_1}{m_1-q_1}}.\) 
By the similar way, we also have
\(a(R_1,{{F}},\Delta ) &=&\frac{\alpha ^R(n(m_1-q_1)+m_1) + \alpha ^Lq_2}{n(m_1-q_1)(m_2-q_2) + m_1(m_2-q_2) + q_2(m_1-q_1)} \\&=& \frac{\frac{\alpha ^R}{m_2-q_2}(n+\frac{m_1}{m_1-q_1})+\frac{\alpha ^L}{m_1-q_1}\frac{q_2}{m_2-q_2}}{n+\frac{m_1}{m_1-q_1}+\frac{q_2}{m_2-q_2}}.\) 
\(\Box \)

Theorem 4.6 
For any DCC set \(B\) , the set
\(\textnormal {PLD}(2,B) := \lbrace \textnormal {pld}_x({{F}},\Delta ) \vert \, \mbox{ $(X,{{F}},\Delta )$ is a foliated triple with $x\in X$ and $\Delta \in B$}\rbrace \) 
satisfies the ascending chain condition (ACC).

Proof Given any non-decreasing sequence \(\lbrace \textnormal {pld}_{x_k}({{F}}_k,\Delta _k)\rbrace _{k=1}^\infty \)  in the set \(\textnormal {PLD}(2,B)\)  where \((X_k,{{F}}_k,\Delta _k)\)  is a germ of foliated triple around \(x_k\)  and \(k \in B\)  for all \(k\) .
We may assume that
\(\textnormal {pld}_{x_k}({{F}}_k,\Delta _k) > 0\) 
for all \(k\) , otherwise the sequence \(\lbrace \textnormal {pld}_{x_k}({{F}}_k,\Delta _k)\rbrace _{k=1}^\infty \)  stabilizes.
Now let \(\varepsilon >0\)  be a number such that \(\textnormal {pld}_{x_k}({{F}}_k,\Delta _k) \ge \varepsilon \)  for all \(k\) .
Since \(B\)  satisfies the descending chain condition, we may assume that \(\min (B\backslash \lbrace 0\rbrace )\ge \varepsilon \) .
Claim We may also assume that the number \(s\)  of irreducible components of \(k\)  is independent of \(k\) .

Proof For ease of our notation, we will drop the subscription \(k\) .
By Lemma REF , the number of irreducible components of exceptional divisors \(E_i\)  which meet \( is bounded by \) N\( for some \) N>0\(.Moreover, by the inequality~(\ref {basic_ineq}), we have \) i=1s ti,jbi2\( where \) s\( is the number of irreducible components of \) . Then
\(2N\ge \sum _{j=1}^r\sum _{i=1}^s t_{i,j}b_i \ge \sum _{i=1}^s\sum _{j=1}^r t_{i,j}\varepsilon \ge \sum _{i=1}^s\varepsilon = s\varepsilon \) 
where the third inequality follows from \(t_{i,j}\ge 1\)  for some \(j\)  for any fixed \(i\) .
Therefore, \(s\le \frac{2N}{\varepsilon }\) .
Hence, after taking a subsequence, we may assume that the number of irreducible components of \(k\)  is independent of \(k\) .

Let \((b_{1,k}\) , \(\ldots \) , \(b_{s,k})\)  be some ordering of coefficients of \(k\) .
After taking a further subsequence, we may assume the sequence \(\lbrace b_{j,k}\rbrace _{k=1}^\infty \)  is non-decreasing for any \(j = 1\) , \(\ldots \) , \(s\) .
If there are infinitely many dual graph \(\Gamma _k\)  belonging to the first case of Lemma REF , then, by Lemma REF  and after taking a further subsequence, we have that \(\textnormal {pld}_{x_k}({{F}}_k,\Delta _k)\)  is non-increasing.
Since it is also non-decreasing, the sequence \(\lbrace \textnormal {pld}_{x_k}({{F}}_k,\Delta _k)\rbrace _{k=1}^\infty \)  stabilizes.
Thus, there are infinitely many dual graphs \(\Gamma _k\)  belonging to the second case of Lemma REF .
After taking a subsequence, we may assume that the dual graphs of \(\lbrace L_1\) , \(\ldots \) , \(L_{\ell _1}\rbrace \)  and \(\lbrace R_1\) , \(\ldots \) , \(R_{\ell _2}\rbrace \)  are fixed, the way how \(B_i\)  intersects \(E_j\)  is fixed, and \(n\)  is sufficiently large.
Since \(b_{j,k}\le 1\) , the limit \(\lim _{k\rightarrow \infty }\, b_{j,k} = \overline{b_j}\)  exists for any \(j = 1\) , \(\ldots \) , \(s\) .
By Lemma REF , we have, for all \(k\) , that
\(\textnormal {pld}_{x_k}({{F}}_k,\Delta _k)\ge \min \left\lbrace \frac{\alpha ^L}{m_1-q_1},\frac{\alpha ^R}{m_2-q_2}\right\rbrace \ge \min \left\lbrace \frac{\overline{\alpha ^L}}{m_1-q_1}, \frac{\overline{\alpha ^R}}{m_2-q_2}\right\rbrace \) 
and
\(\lim _{k\rightarrow \infty }\,\textnormal {pld}_{x_k}({{F}}_k,\Delta _k)= \min \left\lbrace \frac{\overline{\alpha ^L}}{m_1-q_1}, \frac{\overline{\alpha ^R}}{m_2-q_2}\right\rbrace .\) 
Thus, the sequence \(\lbrace \textnormal {pld}_{x_k}({{F}}_k,\Delta _k)\rbrace _{k=1}^\infty \)  stabilizes.
\(\Box \)

Theorem 4.7 
For any DCC set \(B\) , the set
\(\textnormal {MLD}(2,B) := \lbrace \textnormal {mld}_x({{F}},\Delta ) \vert \, \mbox{ $(X,{{F}},\Delta )$ is a foliated triple with $x\in X$ and $\Delta \in B$.}\rbrace \) 
satisfies the ascending chain condition (ACC).

Proof Given any non-decreasing sequence \(\lbrace \textnormal {mld}_{x_k}({{F}}_k,\Delta _k)\rbrace _{k=1}^\infty \)  in the set \(\textnormal {MLD}(2,B)\)  where \((X_k,{{F}}_k,\Delta _k)\)  is a germ of foliated triple around \(x_k\)  and \(k \in B\)  for all \(k\) .
We may assume that
\(\textnormal {pld}_{x_k}({{F}}_k,\Delta _k) > 0\) 
for all \(k\) , otherwise the sequence
\(\lbrace \textnormal {mld}_{x_k}({{F}}_k,\Delta _k)\rbrace _{k=1}^\infty \)  stabilizes.
Now let \(\varepsilon >0\)  be a number such that \(\textnormal {mld}_{x_k}({{F}}_k,\Delta _k) \ge \varepsilon \)  for all \(k\) .
Since \(B\)  satisfies the descending chain condition, we may assume that \(\min (B\backslash \lbrace 0\rbrace )\ge \varepsilon \) .
If there are infinitely many \(k\)  such that \(\textnormal {pld}_{x_k}({{F}}_k,\Delta _k)=\textnormal {mld}_{x_k}({{F}}_k,\Delta _k)\) , then by Theorem REF , we get the sequence \(\lbrace \textnormal {mld}_{x_k}({{F}}_k,\Delta _k)\rbrace _{k=1}^\infty \)  stabilizes.
Thus, by Lemma REF  and REF , after taking a subsequence, we may assume \((X_k,{{F}}_k,\Delta _k)\)  have the same weighted dual graph for the minimal resolution \((Y_k,{{G}}_k,k)\) , the ways how \(B_i\)  intersects \(E_j\)  are the same, and \(\textnormal {pld}_{x_k}({{F}}_k,\Delta _k)>\textnormal {mld}_{x_k}({{F}}_k,\Delta _k)\) .
We may assume the number of irreducible components of \(k\)  is fixed as the claim in the proof of Theorem REF .
Let \((b_{1,k}\) , \(\ldots \) , \(b_{s,k})\)  be some ordering of the coefficients of \(k\) .
Then, by taking a further subsequence, we may assume the sequence \(\lbrace b_{j,k}\rbrace _{k=1}^\infty \)  is non-decreasing for any \(j=1\) , \(\ldots \) , \(s\) .
Since \(\textnormal {pld}_{x_k}({{F}}_k,\Delta _k)>\textnormal {mld}_x({{F}}_k,\Delta _k)\) , there exists an exceptional divisor \(F_k\)  over \(Y_k\)  such that \(a(F_k,{{F}}_k,\Delta _k) = \textnormal {mld}_{x_k}({{F}}_k,\Delta _k)\) .
Then \(F_k\)  is an exceptional divisor of the birational morphism \(\pi _k: Z_k \rightarrow Y_k\)  which is the composition of \(N_k\)  blowups.
Let \(F_{1,k}\) , \(\ldots \) , \(F_{N_k,k} =: F_k\)  be all \(\pi _k\) -exceptional divisors.
We may assume that \(a(F_{j,k},{{F}}_k,\Delta _k)>a(F_k,{{F}}_k,\Delta _k)\)  for all \(j\le N_k-1\) .
Notice that \(-(F_{j,k})^2\ge 2\)  for all \(j \le N_k-1\)  and \([F_k]\)  is not a fork since \(\bigcup _iE_i\cup \bigcup _j F_{j,k}\)  has the simple normal crossing support.
By Lemma REF  (d), the dual graph of \(\lbrace E_i,F_{j,k}\rbrace _{i,j}\)  is a chain.
By Lemma REF , the length of the chain of the dual graph of \(\lbrace E_i,F_{j,k}\rbrace _{i,j}\)  is bounded.
After taking a subsequence, we may assume they have the same dual graphs, and the ways \(B_i\)  intersecting \(E_j\)  and \(F_{j,k}\)  are the same.
Therefore, by Lemma REF , we have that
\(\textnormal {mld}_{x_k}({{F}}_k,\Delta _k) = a(F_k,{{F}}_k,\Delta _k)\ge a(F_{k+1},{{F}}_{k+1},\Delta _{k+1})= \textnormal {mld}_x({{F}}_{k+1},\Delta _{k+1})\) 
for all \(k\) .
This shows that the sequence \(\lbrace \textnormal {mld}_{x_k}({{F}}_k,\Delta _k)\rbrace _{k=1}^\infty \)  stabilizes.
\(\Box \)

Lemma 4.8 
Fix an \(\varepsilon >0\) .
Suppose the sequence \((X_k,{{F}}_k,\Delta _k)\)  with \(\textnormal {pld}_{x_k}({{F}}_k,k)>\textnormal {mld}_{x_k}({{F}}_k,k)\ge \varepsilon \)  and \(b_i\ge \varepsilon \)  have the following properties:

The dual graph for the minimal resolution \((Y_k,{{G}}_k,k)\)  are the same.

The number \(s\)  of irreducible components of \(k\)  are the same.

\((b_{1,k}\) , \(\ldots \) , \(b_{s,k})\)  is some ordering of the coefficients of \(k\) .

\(\lbrace b_{j,k}\rbrace _{k=1}^\infty \)  is non-decreasing for any \(j=1\) , \(\ldots \) , \(s\) .

Then there is a positive integer \(N\)  independent of \(k\)  such that, for each \((Y_k,{{G}}_k,k)\) , there exists a birational morphism \(\pi _k : Z_k \rightarrow Y_k\)  such that

the relative Picard number \(\rho (Z_k/Y_k)\le N\)  and

one of the exceptional divisors on \(Z_k\)  over \(X_k\)  computes the minimal log discrepancy.


Proof To simplify our notation, we will drop the subscript \(k\)  and, for any fixed divisor \(D\) , denote all of the proper transforms of \(D\)  by \(D\) . Since the set \(B\)  satisfies DCC, there is a \(\delta _a>0\)  for any \(a\in {\mathbb {R}}\)  such that \(\sum _{i=1}^sm_i\beta _i-1\ge \delta _a\)  if \(\sum _{i=1}^sm_i\beta _i-a>0\)  for some \(\beta _i\in B\) .
We have seen in the proof of Theorem REF  that the dual graph of \(\lbrace G_n\rbrace =\lbrace E_i,F_j\rbrace \)  is a chain.
This implies that \(\pi \)  is the composition of blowups with center \(p\)  either at the foliation singularities or on two curves at the ends.
We may also assume that \(N\)  is minimal for all \(k\) .
We will proceed induction on \(\sum _{i=1}^s\textnormal {mult}_pB_i\) .
Note that
\(\sum _{i=1}^s\textnormal {mult}_pB_i\le \frac{1}{\varepsilon }\sum _{j=1}^rE_j\le \frac{1}{\varepsilon }\sum _{j=1}^r2 = \frac{2r}{\varepsilon }\) 
by the inequality (REF ) and \(\frac{2r}{\varepsilon }\)  is independent of \(k\) .
Let \(F_1\)  be the exceptional divisor of blowup at \(p\) .
We have following three cases:

Suppose \(p\)  is a smooth foliation point on the curve at the end.
Note that
\(a(F_1,{{F}}, = a(E_r,{{F}}, + 1-\sum _{i=1}^s(\textnormal {mult}_pB_i)b_i.\) 
If all \(B_i\) 's meet \(E_r\)  at \(p\)  transversally and \(\sum _{i=1}^s(\textnormal {mult}_pB_i)b_i\le 1\) , then the exceptional divisors from further blowups have discrepancies at least \(a(F_1,{{F}},\) .
If all \(B_i\) 's meet \(E_r\)  at \(p\)  transversally but \(\sum _{i=1}^s(\textnormal {mult}_pB_i)b_i > 1\) , then
\(\sum _{i=1}^s(\textnormal {mult}_pB_i)b_i -1\ge \delta _1>0\) 
and \(a(F_1,{{F}}, \le a(E_r,{{F}},-\delta _1\le 1-\delta _1\) .
If \(\sum _{i=1}^s\textnormal {mult}_qB_i<\sum _{i=1}^s\textnormal {mult}_pB_i\)  for all \(q\in F_1\) , then we are done by induction.
If not, then the exceptional divisor from blowing up at such \(q\)  has discrepancy \(\le a(F_1,{{F}},-\delta _1\le 1-2\delta _1\) .
Since the minimal log discrepancy \(\textnormal {mld}_x({{F}},\)  is positive, after finitely many blowups, the quantity \(\sum _{i=1}^s\textnormal {mult}_pB_i\)  will strictly decrease.
If some of \(B_i\) 's meet \(E_r\)  at \(p\)  transversally and some do not, then the quantity \(\sum _{i=1}^s\textnormal {mult}_pB_i\)  strictly decreases.
If each \(B_i\)  does not meet \(E_r\)  at \(p\)  transversally, then all \(B_i\) 's meet \(F_1\)  at the reduced singularity \(q\)  on \(F_1\) .
Notice that we have \(\sum _{i=1}^s\textnormal {mult}_qB_i\le \sum _{i=1}^s\textnormal {mult}_pB_i\) .
If it is not an equality, then we are done by induction.
Suppose it is an equality, then we may assume that \(p\)  is contained in two exceptional divisors, say \(E_j\)  and \(E_{j+1}\)  for some \(j\) .

Suppose \(p\)  is contained in two exceptional divisors, say \(E_j\)  and \(E_{j+1}\)  for some \(j\) .
Note that
\(a(F_1,{{F}}, = a(E_j,{{F}},+a(E_{j+1},{{F}},-\sum _{i=1}^s(\textnormal {mult}_pB_i)b_i.\) 
Assume \(a(E_j,{{F}},\le a(E_{j+1},{{F}},\) .
If \(a(F_1,{{F}},\ge a(E_j,{{F}},\) , then all exceptional divisors from further blowups have discrepancies at least \(a(E_j,{{F}},\) .
Thus, we may assume that \(\sigma < a(E_j,{{F}},\) .
Notice that
\(a(E_j,{{F}}, - \sigma = \sum _{i=1}^s(\textnormal {mult}_pB_i)b_i-a(E_{j+1},{{F}}, \ge \delta _\alpha \) 
where \(\alpha = a(E_{j+1},{{F}},\) .
Therefore, \(a(F_1,{{F}}, \le a(E_j,{{F}},-\delta _\alpha \le 1-\delta _\alpha \) .
There are two reduced singularities \(q_1\)  and \(q_2\)  on \(F_1\) .
Note that the quantities \(\sum _{i=1}^s\textnormal {mult}_{q_1}B_i\)  and \(\sum _{i=1}^s\textnormal {mult}_{q_2}B_i\)  are at most \(\sum _{i=1}^s\textnormal {mult}_pB_i\) .
If both are strictly inequalities, then we are done by induction.
If one of them is an equality, say \(q_1\) , then the exceptional divisor \(F_2\)  from blowing up at \(q_1\)  has discrepancy
\(a(F_2,{{F}},\le a(F_1,{{F}}, - \delta _\alpha \le 1-2\delta _\alpha .\) 
Since the minimal log discrepancy \(\textnormal {mld}_x({{F}},\)  is positive, after finitely many blowups, the quantity \(\sum _{i=1}^s\textnormal {mult}_pB_i\)  will strictly decrease.

Suppose \(p\)  is a reduced singularity contained in precisely one exceptional divisor \(E_r\) .
Note that
\(a(F_1,{{F}}, &=& a(E_r,{{F}},-\sum _{i=1}^s(\textnormal {mult}_pB_i)b_i\\&\le & a(E_r,{{F}},-\varepsilon \sum _{i=1}^s\textnormal {mult}_pB_i.\) 
If \(\sum _{i=1}^s\textnormal {mult}_qB_i<\sum _{i=1}^s\textnormal {mult}_pB_i\)  for all \(q\in F_1\) , then we are done by induction.
When \(\sum _{i=1}^s\textnormal {mult}_qB_i = \sum _{i=1}^s\textnormal {mult}_pB_i\)  for some \(q\) , if \(q\)  is either a smooth foliation point or contained in two exceptional divisors, then we are done.
Otherwise, \(q\)  is a reduced singularity contained in precisely one exceptional divisor.
Then the exceptional divisor \(F_2\)  from blowing up at \(q\)  has discrepancy
\(a(F_2,{{F}}, &\le & a(F_1,{{F}}, - \sum _{i=1}^s(\textnormal {mult}_qB_i)b_i \\&\le & a(E_r,{{F}},-2\varepsilon \sum _{i=1}^s\textnormal {mult}_pB_i.\) 
Since the minimal log discrepancy \(\textnormal {mld}_x({{F}},\)  is positive, after finitely many blowups, the quantity \(\sum _{i=1}^s\textnormal {mult}_pB_i\)  will strictly decrease.

\(\Box \)

Remark 4.9 When \(B=\emptyset \) , we have
\(\textnormal {MLD}(2,\emptyset ) = \left\lbrace \frac{1}{n}\vert \, n\in {\mathbb {N}}\right\rbrace \cup \lbrace 0,-\infty \rbrace .\) 


Vanishing theorem for foliations
Definition 5.1 
In Theorem REF , we say a log canonical foliation singularity is good if it is either of the type (1)-(5), or of the type (6) and (7) such that all invariant curves are \((-1)\) -\({{G}}\) -curves and the non-invariant curve \(E\)  satisfying
\(-E^2 \ge \max \lbrace 2p_a(E)-1+\deg \,[E], 2-2p_a(E)\rbrace .\) 

Remark 5.2 All canonical foliation singularities are good log canonical foliation singularities.

Theorem 5.3 
Let \(f : (Y,{{G}}) \rightarrow (X,{{F}})\)  be a proper birational morphism where \((X,{{F}})\)  is a foliated surface with good log canonical foliation singularities and \((Y,{{G}})\)  is a foliated surface with only reduced singularities.
Then \(R^if_*{\mathcal {O}}_Y(K_{{G}})=0\)  for \(i>0\) .

Proof We divide the proof into several steps.

We first consider \((Y,{{G}})\)  is the minimal resolution of \((X,{{F}})\)  with exceptional divisors \(E_1\) , \(\ldots \) , \(E_r\) .
Let \(Z = \sum _{i=1}^ra_iE_i\)  where \(a_i\) 's are non-negative integers.
By the theorem on formal functions, it suffices to show that
\(h^1(Z,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_Z)=0\) 
for any effective (non-zero) divisor \(Z\) .
Let \(A := \sum _{i=1}^ra_i\) .
We will show the vanishing by induction on \(A\) .

When \(A = 1\) , there is exactly one \(i\)  such that \(a_i\)  is positive and equals to 1.
Without loss of generality, we assume \(a_1=1\)  and \(a_i=0\)  for \(i\ge 2\) .
Claim \(\deg (K_{E_1})-K_{{G}}\cdot E_1<0\) .

Proof

If \(E_1\)  is invariant, then we have \(K_{{G}}\cdot E_1\ge -1\)  and
\(\deg (K_{E_1})-K_{{G}}\cdot E_1 = -2-K_{{G}}\cdot E_1\le -1<0.\) 

If \(E_1\)  is non-invariant, then we have
\(\deg (K_{E_1})-K_{{G}}\cdot E_1 &=& 2p_a(E_1)-2+E_1^2 \\&\le & 2p_a(E_1)-2 - (2p_a(E_1)-1+\deg \,[E_1]) \\&=& -1-\deg \,[E_1] < 0.\) 


Thus, \(h^1(E_1,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{E_1}) = h^0(E_1,\omega _{E_1}\otimes {\mathcal {O}}_{E_1}(-K_{{G}})) = 0\) .

Let \(Z_\ell = Z-E_\ell \)  for some \(\ell \)  with positive \(a_\ell \) , which will be determined later.
Then we combine the following two short exact sequences
\({0 [r] & {\mathcal {O}}_Y(-Z) [r] & {\mathcal {O}}_Y(-Z_\ell ) [r] & {\mathcal {O}}_{E_\ell }\otimes {\mathcal {O}}_{Y}(-Z_\ell ) [r] & 0}\) 
and
\({0 [r] & {\mathcal {O}}_Y(-Z_\ell )/{\mathcal {O}}_Y(-Z) [r] & {\mathcal {O}}_Z [r] & {\mathcal {O}}_{Z_\ell }[r] & 0}\) 
into a short exact sequecne
\({0 [r] & {\mathcal {O}}_{E_\ell }\otimes {\mathcal {O}}_{Y}(-Z_\ell ) [r] & {\mathcal {O}}_Z [r] & {\mathcal {O}}_{Z_\ell }[r] & 0}.\) 
Tensoring with \({\mathcal {O}}_Y(K_{{G}})\) , we get
\({0 [r] & {\mathcal {O}}_{E_\ell }\otimes {\mathcal {O}}_Y(K_{{G}}-Z_\ell ) [r] & {\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_Z [r] & {\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{Z_\ell }[r] & 0}.\) 
Now, by induction hypothesis, we have \(h^1(Z_\ell ,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{Z_\ell })=0\) .
Therefore, it suffices to show \(h^1(E_\ell ,{\mathcal {O}}_{E_\ell }\otimes {\mathcal {O}}_Y(K_{{G}}-Z_\ell ))=0\) .
This follows from
\(K_{{G}}\cdot E_\ell -Z_i\cdot E_\ell >\deg (K_{E_\ell }) = K_Y\cdot E_\ell +E_\ell ^2,\) 
which is equivalent to
\((K_{{G}}-K_Y-Z)\cdot E_\ell >0.\) 

Let \(K_{{G}}-K_Y\equiv _f\sum _{i=1}^rb(E_i)E_i\)  for some \(b(E_i)\in {\mathbb {R}}\) .
We will denote \(b(E_i)\)  by \(b_i\)  as well.
Claim \(b(E_i)\le 1\)  for all \(i\) .
Moreover, either \(b(E_i)=1\)  for all \(i\)  or \(b(E_i)<1\)  for all \(i\) .

Proof

If \(E_j\)  is invariant, then we have
\((K_{{G}}-K_Y)\cdot E_j &=& \textnormal {Z}({{G}},E_j)-2+E_j^2+2 \\&=&\textnormal {Z}({{G}},E_j)+E_j^2 \ge \left(\sum _{i=1}^rE_i\right)\cdot E_j.\) 

If \(E_j\)  is non-invariant, then we have
\((K_{{G}}-K_Y)\cdot E_j &=& -E_j^2 + E_j^2+2-2p_a(E_j) \\&> & \deg \,[E_j]+E_j^2 = \left(\sum _{i=1}^rE_i\right)\cdot E_j.\) 

By Lemma REF , we have that \(b_i\le 1\)  for all \(i\) .
Moreover, \(\bigcup _{i=1}^rE_i\)  is connected, we have either \(b_i=1\)  for all \(i\)  or \(b_i<1\)  for all \(i\) .

Claim We have \(b(E_i)\ge 0\)  for all \(i\)  except for the non-invariant \(E_0\)  of (6) and (7) in Theorem REF .

Proof Let \(2D\)  be the sum of all \((-1)\) -\({{G}}\) -curves whose self-intersections are \(-2\) .
Then we have \((K_{{G}}-K_Y)\cdot E_i\le D\cdot E_i\)  for all \(i\) .
Thus, by Lemma REF , we have that \(b_i\ge 0\)  for all \(i\) .
In the case of type (6) and (7), we note that
\((K_{{G}}-K_Y)\cdot E_0\le 2-2p_a(E_0) \le -E_0^2\) 
and
\((K_{{G}}-K_Y)\cdot E_j = 1+E_j^2\le -1 = -E_0\cdot E_j\) 
for any invariant curve \(E_j\) .
Then, by Lemma REF , we have \(b(E_j)\ge 0\)  for all invariant curves \(E_j\) .



Let \(W = \sum _{i=1}^rc_iE_i\)  be an \(f\) -anti-ample effective divisor where \(c_i>0\)  for all \(i\) .
We have following two cases:

If \(a_i-b_i\)  is positive for some \(i\) , then we define \(\alpha \in {\mathbb {R}}_{\ge 0}\)  to be the maximal number such that \(\alpha (a_i-b_i)\le c_i\)  for all \(i\) .
Then there is in index \(\ell \)  such that \(c_\ell -\alpha (a_\ell -b_\ell )=0\) , and therefore
\(\big (W+\alpha (K_{{G}}-K_Y-Z)\big )\cdot E_\ell = \left(\sum _{i=1}^r \big (c_i-\alpha (a_i-b_i)\big )E_i\right)\cdot E_\ell \ge 0.\) 
Hence, we have
\((K_{{G}}-K_Y-Z)\cdot E_\ell \ge \frac{-1}{\alpha }W\cdot E_\ell >0.\) 

If \(a_i-b_i\le 0\)  for all \(i\) , then \(a_i\le b_i\le 1\)  for all \(i\) .
Thus, \(b_i=1\)  for all \(i\) , otherwise, we have \(b_i<1\)  for all \(i\) , and then \(a_i=0\)  for all \(i\)  since \(a_i\) 's are non-negative integers.


Suppose \(\bigcup _{i=1}^rE_i\)  forms a dual graph for a canonical foliation singularity.
Then we have \(b(E_i)\ge 0\)  for all \(i\) .
If \(b_i<1\)  for all \(i\) , then, from \(c_\ell -\alpha (a_\ell -b_\ell )=0\) , we have \(a_\ell = b_\ell +\frac{c_\ell }{\alpha }>0\) .
If \(b_i=1\)  for all \(i\) , then \(a_i\le 1\)  for all \(i\)  and \(\bigcup _{i=1}^rE_i\)  is an elliptic Gorenstein leaf \(\Gamma \) .
By [1]}, we have \({\mathcal {O}}_\Gamma (K_{{G}})\)  is not torsion and has degree 0.
Also \(\Gamma \)  is Cohen-Macaulay with trivial dualizing sheaf, then by Serre duality, we have
\(h^1(\Gamma ,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_\Gamma ) = h^0(\Gamma ,{\mathcal {O}}_Y(-K_{{G}})\otimes {\mathcal {O}}_\Gamma ) = 0.\) 
Thus, we have shown the case when \(a_i=1\)  for all \(i\) .
Therefore, when \(a_i=0\)  for some \(i\) , we have an index \(j\)  such that
\((K_{{G}}-K_Y-Z)\cdot E_j = \left(\sum _i(b_i-a_i)E_i\right)\cdot E_j> 0.\) 

Now suppose \(\bigcup _{i=1}^rE_i\)  forms a dual graph for a non-canonical foliation singularity.
Note that we have \(b_i<1\)  for all \(i\) .
If \(E_\ell \)  in step (REF ) is invariant, then from \(c_\ell -\alpha (a_\ell -b_\ell )=0\) , we have \(a_\ell = b_\ell +\frac{c_\ell }{\alpha }>0\) .
Thus, it suffices to consider the case when \(E_\ell \)  is non-invariant and \(a_\ell = 0\) .
Then the support of \(Z = \sum _iZ_i = \sum _ia_iE_i\)  is the disjoint union of \((-1)\) -\({{G}}\) -curves \(E_i\)  where \(Z_i = a_iE_i\) .
Note that
\(H^1(Z,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_Z)=\bigoplus _i H^1(Z_i,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{Z_i}).\) 
Since the contraction of \(E_i\)  introduces a terminal foliation singularity, we have \(h^1(Z_i,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_{Z_i}) = 0\)  for all \(i\) , and thus \(h^1(Z,{\mathcal {O}}_Y(K_{{G}})\otimes {\mathcal {O}}_Z)=0\) .

Now we consider the general resolution.
Note that any general resolution \(f : (Y,{{G}}) \rightarrow (X,{{F}})\)  factors through the minimal resolution \(h : (Z,{{H}}) \rightarrow (X,{{F}})\) .
Let \(f = h\circ g\) .
So \(g\)  is the composition of blowups.
We have shown that \(R^1h_*{\mathcal {O}}_Z(K_{{H}}) = 0\) .
Notice that \(g_*{\mathcal {O}}_Y(K_{{G}}) = {\mathcal {O}}_Z(K_{{H}})\) .
By Grothendieck spectral sequence, we have the following exact sequence:
\({0 [r] & R^1h_*(g_*{\mathcal {O}}_Y(K_{{G}})) [r] & R^1f_*{\mathcal {O}}_Y(K_{{G}}) [r] & h_*R^1g_*{\mathcal {O}}_Y(K_{{G}}).}\) 
Since \(R^1h_*(g_*{\mathcal {O}}_Y(K_{{G}})) = R^1h_*{\mathcal {O}}_Z(K_{{H}})=0\)  and \(R^1g_*{\mathcal {O}}_Y(K_{{G}})=0\)  by Lemma REF , we have \(R^1f_*{\mathcal {O}}_Y(K_{{G}})=0\) .
\(\Box \)


Lemma 5.4 
Let \((X,{{F}},p)\)  is a germ of a foliated surface with at worst reduced singularity at \(p\) .
Let \(\pi : (Y,E) \rightarrow (X,p)\)  be a blowup at \(p\)  and \({{G}}\)  be the pullback foliation of \({{F}}\) .
Then we have \(R^1\pi _*{\mathcal {O}}_Y(K_{{G}}) = 0\) .

Proof Note that \(K_{{G}}= \pi ^*K_{{F}}+ a(E)E\)  where \(a(E)\)  is either 0 or 1.
Also, since \(p\)  is a smooth point of \(X\) , we have \(R^1\pi _*{\mathcal {O}}_Y=0\) .
If \(a(E)=0\) , then, by the projection formula, we have
\(R^1\pi _*{\mathcal {O}}_Y(K_{{G}}) = R^1\pi _*\pi ^*{\mathcal {O}}_X(K_{{F}}) = R^1\pi _*{\mathcal {O}}_Y\otimes {\mathcal {O}}_X(K_{{F}})=0.\) 
If \(a(E)=1\) , then we consider the following short exact sequence:
\({0 [r] & \pi ^*{\mathcal {O}}_X(K_{{F}}) [r] & {\mathcal {O}}_Y(K_{{G}}) [r] & {\mathcal {O}}_E(E) [r] & 0}\) 
Pushing forward via \(\pi \) , we obtain the exact sequence
\({0=R^1\pi _*\pi ^*{\mathcal {O}}_X(K_{{F}}) [r] & R^1\pi _*{\mathcal {O}}_Y(K_{{G}}) [r] & R^1\pi _*{\mathcal {O}}_E(E) [r] & 0}.\) 
Since \(R^1\pi _*{\mathcal {O}}_E(E) = H^1({\mathbb {P}}^1,{\mathcal {O}}_{{\mathbb {P}}^1}(-1)) = 0\) , we have \(R^1\pi _*{\mathcal {O}}_Y(K_{{G}}) = 0\) .
\(\Box \)

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